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The image you've uploaded contains a math problem about finding the area of a polygon using its vertices. It looks like the problem is asking for the formula to compute the area based on the vertices in counterclockwise order, and there's a specific set of choices provided for the formula. Also, part (c) of the question involves calculating the area of a pentagon using specific vertex coordinates.

Let's first break down the question:

### Part (b)
This part describes a polygon with vertices \( (x_1, y_1), (x_2, y_2), \dots, (x_n, y_n) \) in counterclockwise order. It asks you to find the area of the polygon using the following formula:

\[
A = \frac{1}{2} \left[ (x_1 y_2 - x_2 y_1) + (x_2 y_3 - x_3 y_2) + \cdots + (x_{n-1} y_n - x_n y_{n-1}) + (x_n y_1 - x_1 y_n) \right]
\]

This formula is known as the **shoelace theorem** (or Gauss's area formula). The formula sums up the terms formed by the products of the coordinates, following the counterclockwise order of the vertices.

### Part (c)
For part (c), you are tasked with calculating the area of a pentagon given the vertices \( (0, 0), (1, 3), (1, 2), (0, 1), (-3, 1) \). You would plug these coordinates into the shoelace formula to compute the area.

Let me know if you'd like me to go step-by-step to calculate the area of the pentagon or if you have further questions about the theory!

If the vertices of a polygon, in counterclockwise order, are (x1, y1), (x2, y2), ..., (xn, yn), find the area of the polygon.

Learn how to calculate the area of a polygon using the Shoelace Theorem. The formula involves using the coordinates of the polygon's vertices in counterclockwise order. Example problem: find the area of a pentagon with specific vertex coordinates.

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